Question

The Setup: A piece of wire of length 10 cm is cut into two (not necessarily equal length) pieces. One piece of length x cm is made into a circle and the rest is made into a square.

1. Express the sum of the areas of the square and circle as a function of x

2. For what x-value is maximum area achieved?

3. For maximum area, how should the wire be bent?

4. Find the critical values of your function f[x].

5. Now give the radius and side length of the circle and square
for which area is

minimized.

6. A curious property of the circle and square that maximize or
minimize volume follows. For the x-values you computed that
minimized and maximized the combined areas of the circle and the
square, have Mathematica verify the following for you:

Length of wire used for the square/Length of wire used for the
circle = area of square/area of circle

Use exact values and the simplify command to verify that both
ratios are the same.

7. You might wonder if there are are other values of x for
which,

Length of wire used for the square/Length of wire used for the
circle = area of square/area of circle

Use the Solve[ ] command in Mathematica by solving this proportion
for x, where x is the circumference of the circle as before. What
do you notice about the solutions to this equation and the values
of x that minimized and maximized the combined areas?

Answer #1

A piece of wire of length 70 is cut into two pieces. One piece
is bent into a square and the other is bent into a circle. If the
sum of the areas enclosed by each part is a minimum, what is the
length of each part?
To minimize the combined area, the wire should be cut so that a
length of ____ is used for the circle and a length of ______is used
for the square.
(Round to the...

A piece of wire of length
6161
is cut, and the resulting two pieces are formed to make a
circle and a square. Where should the wire be cut to (a) minimize
and (b) maximize the combined area of the circle and the
square?
(a) To minimize the combined area, the wire should be cut so
that a length of
___
is used for the circle and a length of
___
is used for the square.

A piece of wire 10 m long is cut into two pieces. One piece is
bent into a square and the other is bent into a circle. (a) How
much wire should be used for the square in order to maximize the
total area? & How much wire should be used for the square in
order to minimize the total area?

A wire is to be cut into two pieces. One piece will be bent into
an equilateral triangle, and the other piece will be bent into a
circle. If the total area enclosed by the two pieces is to be 100 m
^2 , what is the minimum length of wire that can be used? (Use
decimal notation. Give your answer to one decimal place.)
L min = ? cm
What is the maximum length of wire that can be...

A piece of wire of length 55 is cut, and the resulting two
pieces are formed to make a circle and a square. Where should the
wire be cut to (a) minimize and (b) maximize the combined area of
the circle and the square?

A piece of wire of length
5353
is cut, and the resulting two pieces are formed to make a
circle and a square. Where should the wire be cut to (a) minimize
and (b) maximize the combined area of the circle and the
square?

100 cm long wire is cut into two pieces. One of the pieces is
bent into a circle, but square is made from the other wire piece.
How is the 100 cm wire supposed to be cut so that the total
area of the circle and the square will be
(a) the biggest.
(b) the smallest.

100 cm long wire is cut into two pieces. One of the pieces is
bent into a circle, but square is made from the other wire piece.
How is the 100 cm wire supposed to be cut so that the total
area of the circle and the square will be
(a) the biggest.
(b) the smallest.

A piece of wire 26 m long is cut into two pieces. One piece is
bent into a square and the other is bent into a circle.
(a) How much wire should be used for the square in order to
maximize the total area?
(b) How much wire should be used for the square in order to
minimize the total area?

A piece of wire 8 m long is cut into two pieces. One piece is
bent into a square and the other is bent into a circle. (Give your
answers correct to two decimal places. ) How much wire should be
used for the circle in order to maximize the total area? m How much
wire should be used for the circle in order to minimize the total
area? m
show all work

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